Commutative Algebra - MTH 423 (for UG + iPhD)
January - April 2017
Instructor : Dr. Anupam Singh
Schedule : Lecture : Tuesday at 9am
Lecture : Thursday at 11pm
Tutorial : Friday at 2pm
Evaluation : Test I - 20%
Mid-sem examination- 30%
Test II- 20%
End-sem examination- 30%
Prerequisite : motivation to learn Algebra
Goal of the course : This topic prepares necessary background for the subject of Algebraic Number Theory and Algebraic Geometry.
Proposed content : Rings, Modules, Localisation, Tensor product, Noetherian and Artinian rings, Integral dependence, Dedekind domains, Noether Normalisation, Hilbert’s Nullstellensatz, Affine algebra over a field.
References :
Steps in Commutative Algebra, R. Y. Sharp
Introduction to Commutative Algebra, Atiyah and Macdonald
Further Reading : One can take up the following subjects from here on: Algebraic Geometry, Algebraic Groups, Elliptic Curves (which is also applied in cryptography), Commutative algebra, Non-commutative associative algebras, Algebraic Number Theory, Computational commutative algebra and Algebraic Geometry, etc.
5 January 2017 - Solving equations, Rings and Modules
6 January 2017 - Examples of rings and modules Assignment I
10 January 2017 - Ideals, prime ideals and maximal ideals
12 January 2017 - (generalised) Euclidean algorithm for polynomials, Example of group ring
13 January 2017 - Tutorials
17 January 2017 - Affine space, Closed sets, Affine algebraic variety
19 January 2017 - Zariski Topology
20 January 2017 - Tutorials
24 January 2017 - Irreducible varieties, Noetherian topological spaces and irreducible components
26 January 2017 - Holiday
27 January 2017 - Tutorials
31 January 2017 - Noetherian Modules, Noetherian rings
2 February 2017 - Hilbert Basis Theorem Assignment II
3 February 2017 - Test I
7 February 2017 - Field of fractions, Localisation of a ring
9 February 2017 - Localisation of a module, examples
10 February 2017 - Tutorial
14 February 2017 - Local property
16 February 2017 - Local Rings
17 February 2017 - Tutorial
23 February 2017 Mid Sem Exam
28 February 2017 - Integral extension, examples
2 March 2017 - Integral closure, integrally closed, examples
3 March 2017 - Tutorial
7 March 2017 - Algebraic independence
9 March 2017 - Noether's Normalisation Lemma
10 March 2017 - Tutorial Assignment III
14 March 2017 - Hilbert's Nullstellensatz (version I)
16 March 2017 - Hilbert's Nullstellensatz (version II), Co-ordinate ring, Affine k-algebra
17 March 2017 - Tutorial Assignment IV
21 March 2017 - variety, morphism of variety, morphism inducing algebra homomorphism at coordinate level
23 March 2017 - Examples
24 March 2017 - Tutorial
28 March 2017 - Holiday
30 March 2017 - Tensor Product of modules
31 March 2017 - Test II
4 April 2017 - Examples of tensor product Assignment V
6 April 2017 - Base change
7 April 2017 - Tutorial
11 April 2017 - Story of Algebraic Groups
13 April 2017 - Guest lecture by Vivek Mallick "Intersection"
14 April 2017 - Holiday
28 April 2017 - End Sem Exam